Question

The equation $x^3-3x+q=0$ will have two roots equal, if the value of $q$ is

• A. $\pm 2$
• B. $\pm 1$
• C. $\pm 3$
• D. $\pm 4$

Answer 1

The correct option is A $\pm 2$

Let the roots are $\alpha$, $\alpha$, $\beta$, therefore
$2\alpha +\beta =0$
or $\beta =-2\alpha$
and ${\alpha }^2\beta =-q$
$\Rightarrow -2{\alpha }^3=-q$
or $a^3=\frac{q}{2}$ ...(i)
Also ${\alpha }^2+\alpha \beta +\alpha \beta =-3$
or $-3\alpha =-3$
or ${\alpha }^2=1$
$\therefore \alpha =\pm 1$
Substituting the value of a in equation (i), we get
$\pm 1=\frac{q}{2}$
or $q=\pm 2$